If α and β are the roots of \(7×2 +12x – 4 = 0\),find the value of \(\frac{αβ}{(α + β)^2}\)
- \( \frac{7}{36}\)
- -\( \frac{36}{7}\)
- \(\frac{36}{7}\)
-
-\( \frac{7}{36}\)
The correct answer is: D
Explanation
The general form of a quadratic equation is:
\(x^2\) -(sum of roots)\(x\) +(product of roots) = 0
\(7x^2+12x-4=0\)
Divide through by 7
=\(x^2+\frac{12}{7}x-\frac{4}{7}=0\)
=\(x^2-(-\frac{12}{7})x+(-\frac{4}{7})=0\)
\(\therefore\) sum of roots = \(-\frac{12}{7}\), and products of roots =\(-\frac{4}{7}\)
α + β = \(-\frac{12}{7}, αβ = -\frac{4}{7}\)
\(\frac{αβ}{(α + β)^2} = \frac{\frac{-4}{7}}{(\frac{-12}{7})^2}\)
=\(\frac{\frac{-4}{7}}{\frac{144}{49}}=-\frac{4}{7}\times\frac{49}{144}\)
\(\therefore - \frac{7}{36}\)